Also notice that if the ambient (∞,1)(\infty,1)-category is in fact an (∞,1)-topos, then every object in there may already be thought of as an “∞-groupoid with geometric structure” (see for instance the discussion at cohesive (∞,1)-topos, but this is true more generally). The relation between the internal groupoid objects then and the objects themselves is (an oid-ification) of that of looping and delooping. Notably for GG any internal group object (externally an ∞-group) the corresponging ordinary object is its delooping object BG\mathbf{B}G, and every pointedconnected object in the (∞,1)(\infty,1)-topos arises this way from an internal group object.

is a (∞,1)-pullback diagram in CC. Here, by a partition S∪S′S \cup S' of [n][n] that share precisely one vertex ss, we mean two subsets SS and S′S' of {0,1,…,n}\{0,1,\ldots,n\} whose union is {0,1,…,n}\{0,1,\ldots,n\} and whose intersection is the singleton {s}\{s\}. The linear order on [n][n] then restricts to the linear order on SS and S′S'.

Remark

Definition

A groupoid object A:Δop→CA : \Delta^{op} \to C is the Cech nerve of a morphism A0→BA_0 \to B if AA is the restriction of an augmented simplicial object A+:Δaop→CA^+ : \Delta^{op}_a \to C with A0+→A−1+A^+_0 \to A^+_{-1} as the morphism A0→BA_0 \to B, such that the sub-diagram

Properties

Equivalent characterizations

We state in prop. 1 below a list of equivalent conditions that characterize a simplicial object in an (∞,1)-category as a groupoid object. This uses the following basic notions, which we review here for convenience.

Proof

In one direction: the limit is the terminal object in the cone category, and so is preserved by equivalences of cone categories. (This direction appears as (Lurie, prop. 4.1.1.8)). Conversely, the limits is the object representing cones, and hence an equivalence of limits induces an equivalence of cone categories.

Proposition

Let 𝒞\mathcal{C} be an (∞,1)(\infty,1)-category incarnated explicitly as a quasi-category. Then a simplicial object in𝒞\mathcal{C} is a groupoid object if the following equivalent conditions hold.

For every n≥2n \geq 2 and every 0≤i≤n0 \leq i \leq n, the morphism 𝒞/X[Δn]→𝒞/X[Λin]\mathcal{C}_{/X[\Delta^n]} \to \mathcal{C}_{/X[\Lambda^n_i]} is an weak equivalence in the model structure for quasi-categories

(…)

Using remark 3 this means equivalently that the simplicial object X•X_\bullet is a groupoid precisely if the following